Bias Correction in VHM Estimation

Why Observed Risk Means Need a Correction

In the Bühlmann-Straub model, observed experience for each risk reflects two sources of variation. Risks differ in their underlying expected outcomes, and each risk's observed experience also fluctuates around its own expected outcome. Consequently, the spread of observed risk means is generally larger than the spread of the underlying risk means alone. Treating the entire observed spread as VHM (variance of hypothetical means) therefore overstates VHM.

The correction depends on how risk means are summarized and on the exposures supporting them. With equal exposures, the process contribution has a simple form. With unequal exposures, a risk mean based on less exposure has greater process variance, so the correction must account for the exposure pattern. This tutorial derives both the unweighted and exposure-weighted corrections.

The exposure-scaled variance assumption means that more exposure reduces the process variance of a risk's experience mean. Conditional independence gives \(\operatorname{Var}(\bar X_i\mid\mu_i)=\text{EPV}/w_i\). The observed risk mean thus contains both its latent risk mean and sampling fluctuation around that mean.

Unweighted Sample Variance: Derivation of the Correction

An unweighted estimator gives every risk mean equal influence, regardless of its exposure. Define the ordinary sample mean of the risk means as \(\bar X_{\cdot}=r^{-1}\sum_i\bar X_i\), and their sample variance as \(S^2_{\bar X}=(r-1)^{-1}\sum_i(\bar X_i-\bar X_{\cdot})^2\). The same risk-level variance decomposition applies whether exposures are equal or unequal; the exposure pattern determines the process contribution.

To see why, write \(\bar X_i=\mu_i+\varepsilon_i\), where \(\operatorname{E}[\varepsilon_i\mid\mu_i]=0\) and \(\operatorname{Var}(\varepsilon_i)=\text{EPV}/w_i\). The risk means are independent across risks and have common expectation \(m\), but their variances are \(\text{VHM}+\text{EPV}/w_i\). For independent observations with common expectation, the expected ordinary sample variance is the average of their variances. Specifically, if their variances are \(\sigma_i^2\), then \(\operatorname{E}[\sum_i(\bar X_i-\bar X_{\cdot})^2]=\sum_i\sigma_i^2-r\operatorname{Var}(\bar X_{\cdot})\), and \(\operatorname{Var}(\bar X_{\cdot})=r^{-2}\sum_i\sigma_i^2\). Dividing by \(r-1\) gives \(r^{-1}\sum_i\sigma_i^2\). Substituting \(\sigma_i^2=\text{VHM}+\text{EPV}/w_i\) yields the result.

Unequal Exposures: Two Estimators, Two Corrections

When exposures differ, there are two natural ways to summarize the spread of risk means. An unweighted sample variance gives each risk equal weight and uses the average reciprocal exposure correction derived above. An exposure-weighted dispersion gives more influence to more-exposed risks and has a different denominator and correction. These estimators should not be mixed: the correction must match the dispersion statistic used.

For justification, write the weighted dispersion in quadratic form as \(\sum_i w_i\bar X_i^2-(\sum_i w_i\bar X_i)^2/W\). The \(\text{VHM}\) part of its expectation is \(\text{VHM}(W-\sum_iw_i^2/W)=\text{VHM}\,D\). The process part is \(\sum_iw_i(\text{EPV}/w_i)-W^{-1}\sum_iw_i^2(\text{EPV}/w_i)=r\text{EPV}-\text{EPV}=(r-1)\text{EPV}\). Subtracting that contribution and dividing by \(D\) removes the expected process variation while putting the remaining dispersion on the VHM scale. If exposures are all \(w\), then \(D=(r-1)w\), and the correction to \(Q_w/D\) reduces to \(\text{EPV}/w\), agreeing with the balanced result.

Interpreting the Estimate and Avoiding Exam Traps

These corrections remove expected process variation; they do not guarantee that every finite-sample estimate will be positive. A corrected method-of-moments estimate can be negative if the observed dispersion is smaller than the estimated process contribution. That result indicates insufficient observed between-risk variation to support a positive raw estimate under this calculation. If a procedure imposes a nonnegative constraint and replaces a negative estimate by zero, that constrained value is not the same unbiased method-of-moments estimate.

The main exam trap is using a correction that belongs to a different statistic. For the unweighted sample variance, subtract \(\text{EPV}\) times the average of \(1/w_i\). For the weighted dispersion divided by \(D\), subtract \((r-1)\text{EPV}/D\). In balanced data, both give \(\text{EPV}/w\); this agreement is a useful check, but does not make the unequal-exposure formulas interchangeable.